Final answer:
The question demonstrates a vector identity involving the dot product and the cross product, where the sum of their squares equals the square of the original vector's magnitude.
Step-by-step explanation:
The question asks to show that for any vector b, the equation b² = (u.b)² + (u x b)² holds, where u is a fixed unit vector. This equation is derived from the properties of the dot product and cross product in vector mathematics. The dot product of a vector with itself gives the square of its magnitude, and the cross product gives a vector perpendicular to the original vectors. Therefore, the magnitude squared of the cross product represents the square of the component of the vector b that is perpendicular to u. Hence, the sum of the squares of the dot product and the magnitude of the cross product equals the square of the magnitude of b, which is the parallelogram law in vector mathematics.